Cisinski's converse conjecture for the rational motivic spectrum

Let $SH(\mathbb{F}_q)^c_\mathbb{Q}$ be the compact rational stable motivic homotopy category over $\mathbb{F}_q$, and let $\operatorname{Spc}$ denote its tensor-triangular spectrum. Cisinski's conjecture. If

$$\operatorname{Spc}(SH(\mathbb{F}_q)^c_\mathbb{Q}) \cong \mathbb{Q},$$

then the Beilinson–Parshin conjecture holds and rational equivalence agrees with numerical equivalence. This is presented as a converse to the implication that these motivic conjectures determine the tensor-triangular spectrum; the source does not establish the converse or indicate that it has been resolved.